End reductions, fundamental groups, and covering spaces of irreducible open 3–manifolds

Myers, Robert

doi:10.2140/gt.2005.9.971

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End reductions, fundamental groups, and covering spaces of irreducible open 3–manifolds

Robert Myers

Geometry & Topology 9 (2005) 971–990

DOI: 10.2140/gt.2005.9.971

arXiv: math.GT/0407172

Abstract

Suppose $M$ is a connected, open, orientable, irreducible 3–manifold which is not homeomorphic to $ℝ$ . Given a compact 3–manifold $J$ in $M$ which satisfies certain conditions, Brin and Thickstun have associated to it an open neighborhood $V$ called an end reduction of $M$ at $J$ . It has some useful properties which allow one to extend to $M$ various results known to hold for the more restrictive class of eventually end irreducible open 3–manifolds.

In this paper we explore the relationship of $V$ and $M$ with regard to their fundamental groups and their covering spaces. In particular we give conditions under which the inclusion induced homomorphism on fundamental groups is an isomorphism. We also show that if $M$ has universal covering space homeomorphic to $ℝ$ , then so does $V$ .

This work was motivated by a conjecture of Freedman (later disproved by Freedman and Gabai) on knots in $M$ which are covered by a standard set of lines in $ℝ$ .

Keywords

3–manifold, end reduction, covering space

Mathematical Subject Classification 2000

Primary: 57M10

Secondary: 57N10, 57M27

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Publication

Received: 14 July 2004

Revised: 18 May 2005

Accepted: 18 May 2005

Published: 29 May 2005

Proposed: David Gabai

Seconded: Walter Neumann, Cameron Gordon

Authors

Robert Myers
Department of Mathematics Oklahoma State University Stillwater Oklahoma 74078 USA

Volume 9, issue 2 (2005)